A qualitative analysis of solutions to microelectromechanical systems with curvature and nonlinear permittivity profile

An investigation of qualitative properties of solutions to microelectromechanical systems with nonlinear permittivity profile is established. The considered problem is described by a quasilinear evolution equation for the displacement of a membrane, coupled with an elliptic moving boundary problem for the electrostatic potential between the moving membrane and a rigid ground plate. The system is shown to be well-posed locally in time for arbitrary values λ of the applied voltage. For small values of λ the solution exists even globally in time such that the membrane never touches down on the ground plate. In addition to those fundamental results concerning existence and uniqueness of solutions, sufficient conditions are specified which guarantee non-positivity of the membrane's displacement on the one hand and the occurrence of a singularity after a finite time T on the other hand.     

Non-positivity and oscillations of solutions of nonlinear stochastic difference equations with state-dependent noise

Non-positivity and oscillations of nonlinear stochastic difference equations are discussed to characterize the long-term behaviour of their solutions in almost sure sense. Moreover, we present results on non-positivity with probability less than one and give numerous examples satisfying our main hypotheses.